KdV and mKdV Equations of Higher-orders

  • Abdul-Majid Wazwaz
Part of the Nonlinear Physical Science book series (NPS)


It is well known that the third order KdV equation is the generic model for studying weakly nonlinear waves. The equation models surface waves with small amplitude and long wavelength on shallow water. The KdV equation involves a balance between weak nonlinearity and linear dispersion. The KdV equation is completely integrable and the collision between solitary waves is elastic, which means that the solitons retain original identity after collision.


Soliton Solution mKdV Equation Kawahara Equation Singular Soliton Solution Single Soliton Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution and Inverse Scattering, Cambridge University Press, Cambridge (1991).zbMATHCrossRefGoogle Scholar
  2. 2.
    P.J. Caudrey, R.K. Dodd, and J.D. Gibbon, A new hierarchy of Korteweg-de Vries equations, Proc. Roy. Soc. Lond. A 351, 407–422, (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulation, 43(1), 13–27, (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdVtype bilinear equations, J. Math. Phys., 28(8), 1732–1742, (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge (2004).zbMATHCrossRefGoogle Scholar
  6. 6.
    R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A 85,407–415, (1981).CrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Ito, An extension of nonlinear evolution equations of the KdV (mKdV) type to higher orders, J. Phys. Soc. Japan, 49(2), 771–778, (1980).CrossRefMathSciNetGoogle Scholar
  8. 8.
    D.J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class Ψ xxx+6 x+6=λΨ, Stud. Appl. Math., 62, 189–216, (1980).zbMATHMathSciNetGoogle Scholar
  9. 9.
    B.A. Kuperschmidt, A super KdV equation: an integrable system, Phys. Lett. A 102, 213–215, (1984).CrossRefMathSciNetGoogle Scholar
  10. 10.
    P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21, 467–490, (1968).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Nakamura, Simple explode-decay mode solutions of a certain one-space dimensional nonlinear evolution equations, J. Phys. Soc. Japan, 33(5), 1456–1458, (1972).CrossRefGoogle Scholar
  12. 12.
    K. Sawada and T. Kotera, A method for finding N-solition solutions of the K.d.V. equation and the K.d.V.-like equation, Progr. Theoret. Phys., 51, 1355–1367, (1974).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. W. Tam, W.X. Ma, X.B. Hu, and D.L. Wang, The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited, J. Phys. Soc. Japan, 69(1), 45–52, (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A.M. Wazwaz, The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves, Appl. Math. Comput, 201, 489–503, (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A.M. Wazwaz, Multiple-solution solutions of two extended model equations for shallow water waves, Appl. Math. Comput., 201, 790–799, (2008).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Abdul-Majid Wazwaz
    • 1
  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA

Personalised recommendations